F(x)=x4-2x2+3

f(x) = x^4 − 2x^2 + 3 can be rewritten as f(x) = (x^2 − 1)^2 + 2 Key properties: - Domain: all real numbers. - Range: [2, ∞) since (x^2 − 1)^2 ≥ 0. - It’s an even function (f(-x) = f(x)). Critical points (derivative): - f'(x) = 4x^3 − 4x = 4x(x^2 − 1) - Critical points at x = −1, 0, 1 Second derivative: - f''(x) = 12x^2 − 4 - f''(−1) = 8 > 0 ⇒ local min at x = −1, f(−1) = 2 - f''(0) = −4 < 0 ⇒ local max at x = 0, f(0) = 3 - f''(1) = 8 > 0 ⇒ local min at x = 1, f(1) = 2 Global behavior: - Global minima at x = ±1 with value 2 (so the minimum of f is 2). - f(x) → ∞ as |x| → ∞ (unbounded above). No real zeros since f(x) ≥ 2 > 0. If you want, I can sketch the graph, or solve a specific optimization problem (e.g., find where f is increasing/decreasing, or find extrema on an interval). What would you like to do next?